On the Minimum Error Correction Problem for Haplotype Assembly in Diploid and Polyploid Genomes

Abstract

In diploid genomes, haplotype assembly is the computational problem of reconstructing the two parental copies, called haplotypes, of each chromosome starting from sequencing reads, called fragments, possibly affected by sequencing errors. Minimum error correction (MEC) is a prominent computational problem for haplotype assembly and, given a set of fragments, aims at reconstructing the two haplotypes by applying the minimum number of base corrections. MEC is computationally hard to solve, but some approximation-based or fixed-parameter approaches have been proved capable of obtaining accurate results on real data. In this work, we expand the current characterization of the computational complexity of MEC from the approximation and the fixed-parameter tractability point of view. In particular, we show that MEC is not approximable within a constant factor, whereas it is approximable within a logarithmic factor in the size of the input. Furthermore, we answer open questions on the fixed-parameter tractability for parameters of classical or practical interest: the total number of corrections and the fragment length. In addition, we present a direct 2-approximation algorithm for a variant of the problem that has also been applied in the framework of clustering data. Finally, since polyploid genomes, such as those of plants and fishes, are composed of more than two copies of the chromosomes, we introduce a novel formulation of MEC, namely the k-ploid MEC problem, that extends the traditional problem to deal with polyploid genomes. We show that the novel formulation is still both computationally hard and hard to approximate. Nonetheless, from the parameterized point of view, we prove that the problem is tractable for parameters of practical interest such as the number of haplotypes and the coverage, or the number of haplotypes and the fragment length.

A Graph Auto-Encoder for Haplotype Assembly and Viral Quasispecies Reconstruction

“…The vast majority of existing haplotype assembly methods attempt to remove the aforementioned ambiguity by altering or even discarding the data, leading to minimum SNP removal (Lancia 2001), maximum fragments cut (Duitama 2010), and minimum error correction (MEC) score optimization criteria. Majority of haplotype assembly methods developed in recent years are focused on optimizing the MEC score, i.e., determining the smallest possible number of nucleotides in sequencing reads that should be altered such that the resulting dataset is consistent with having originated from k haplotypes (k denotes the ploidy of an organism) (Xie 2016;Pirola 2015;Kuleshov 2014;Patterson 2015;Bonizzoni 2016). These include the branch-and-bound scheme (Wang 2005), an integer linear programming formulation in (Chen 2013), and a dynamic programming framework in (Kuleshov 2014).…”

Resolving Multicopy Duplications de novo Using Polyploid Phasing

“…Beginning with Hapcompass [1], there has been some work on polyploid phasing using algorithms based on branch-and-extend [5], belief propagation [32] and semi-definite programming [14]. In a recent theoretical work [7], the hardness of optimizing the MEC for S > 2 has also been proven, indicating that algorithms for this problem need to be necessarily approximate or tailored to some assumptions. A major drawback of existing works is that they consider only S = 3 , 4 and none have been developed, optimized, or tested for the high ploidy that is encountered in segmental duplications, where S can be potentially larger than 10, and to the low error-rate in Illumina sequencers.…”

Sparse Tensor Decomposition for Haplotype Assembly of Diploids and Polyploids

“…For this reason, the vast majority of haplotype assembly techniques attempts to remove the aforementioned ambiguities by either discarding or altering sequencing data; this has led to the minimum fragment removal, minimum SNP removal [26], maximum fragments cut [16], and minimum error correction formulations of the assembly problem [29]. Most of the recent haplotype assembly methods (see, e.g., [7,25,31,32,40]) focus on the minimum error correction (MEC) formulation where the goal is to nd the smallest number of nucleotides in reads that need to be changed so that any read partitioning ambiguities would be resolved. It has been shown that nding optimal solution to the MEC formulation of the haplotype assembly problem is NP-hard [7,10,26].…”

“…Most of the recent haplotype assembly methods (see, e.g., [7,25,31,32,40]) focus on the minimum error correction (MEC) formulation where the goal is to nd the smallest number of nucleotides in reads that need to be changed so that any read partitioning ambiguities would be resolved. It has been shown that nding optimal solution to the MEC formulation of the haplotype assembly problem is NP-hard [7,10,26]. In [39], the authors used a branch-and-bound scheme to minimize the MEC objective over the space of reads; to reduce the search space, they relied on a bound on the objective obtained by a random partition of the reads.…”